Contact Geometry, Fano Orbifolds, and Einstein Metrics

抽象DMS-0203219 Boyer和Galicki教授建议研究几个几何和拓扑学项目。所有项目的目的都是研究黎曼几何中的基本问题，主要集中在两个方面：Fano簇上分支丛的接触几何和这种空间上某些特殊的(即，爱因斯坦的，正的Ricci曲率)度量的存在性。这里提出的问题深深植根于主要研究者的早期工作，他们利用了Sasakian-Einstein空间的接触几何与两种Kaehler几何之间的基本关系，即具有正标量曲率的Kaehler-Einstein或双曲度量的q-阶乘Fano簇，以及具有Kaehler-Ricci平坦度量的Calabi-Yau流形。主要研究人员的最新工作在这类结构的研究上取得了重大突破，主要研究人员利用Demailly和Kollar的最新结果，能够构造出5维紧致爱因斯坦流形的新例子，以及有理同调7球面族上的许多正爱因斯坦度量。主要研究人员使用的技术借鉴了几个不同的领域：Mori理论和交集理论的代数几何，Calabi猜想的分析，以及孤立超曲面奇异链环的经典微分拓扑。这些方法可以扩展得更远，方向是不变的。在这方面突出的一个例子是奇异球体上爱因斯坦度规的存在问题，这是本提案的主要目标之一。更广泛地说，主要研究人员想要解决关于5维和7维紧致Sasakian-Einstein流形的几个分类问题。这两个维是重要的，有两个不同的原因。鉴于前面的工作，高维的例子可以使用连接结构来构造。同时，这两个奇数维度似乎在超弦理论中扮演着特殊的角色。在弦和M-理论的最新发展的背景下，主要研究人员还建议研究与4维的自对偶爱因斯坦度量以及7维和8维的例外完整度量有关的一些相关问题。本项目旨在加深对某些重要几何类型的数学的理解。虽然这一努力并不是由技术进步直接推动的，但数学史充分表明，今天的纯数学往往成为明天的应用数学。事实上，这个项目中正在考虑的数学目前正被用于初级粒子物理领域，更具体地说，是试图理解对宇宙基本力的统一描述。

ABSTRACT DMS - 0203219.Professors Boyer and Galicki propose to investigate several projects in geometry and topology. The objective of all the projects is to study fundamental questions in Riemannian Geometry with two main focal points: Contact Geometry oforbifold bundles over Fano varieties and the existence of somespecial (i.e., Einstein, positive Ricci curvature) metrics on such spaces. The questions and problems proposed here are deeply rooted in theprincipal investigators' earlier work which exploited a fundamental relationship between contact geometry of Sasakian-Einstein spaces and two kinds of Kaehler geometry, namely Q-factorial Fano varieties with Kaehler-Einstein orbifold metrics with positive scalar curvature, and Calabi-Yau manifolds with their Kaehler Ricci-flat metrics. The most recent work of the principal investigators has led to an important breakthrough in the study of such structures when,using recent results of Demailly and Kollar, the principal investigatorswere able to construct new examples of compact Einstein manifolds in dimension 5 as well as many positive Einstein metrics on families of rationalhomology 7-spheres. The techniques used by the principal investigators borrow fromseveral different fields; the algebraic geometry of Mori theory and intersectiontheory, the analysis of the Calabi Conjecture, and finally theclassical differential topology of links of isolated hypersurfacesingularities. These methods can be extended much further and invarious directions. An example that stands out in this respect is the question of the existence of Einstein metrics on exotic spheres, which is one of the main objectives of this proposal. More generally the principal investigators want to address several classification problems concerning compact Sasakian-Einstein manifolds in dimensions 5 and 7. These two dimensions are important for two separate reasons. In view of earlier work higher dimensional examples can be constructed using the join construction. At the same time these two odd dimensions appear to play special role in Superstring Theory. In the context of recent developements inString and M-Theory the principal investigators also propose toinvestigate some related problems concerning self-dual Einstein metrics in dimension 4 and exceptional holonomy metrics in dimension 7 and 8.This project is intended to further the understanding of the mathematicsof certain important types of geometry. While this endeavor is notdirectly motivated by advances in technology, the history of mathematicsadequately demonstrates that today's pure mathematics often becomestomorrow's applied mathematics. Indeed the mathematics being considered inthis project is currently being used in the field of Elementary ParticlePhysics, more specifically, in the attempts at understanding a unifieddescription of the fundamental forces of the universe.-